Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 219 



Mr. Ivory states tlie following equations of condition: 



e = a so — tn 

 e = ax — m 

 &c. = &c. 

 He thence deduces the following value of w: 



X = ., + , and those of e e are 



a'S.am a'S.ae 

 e = -m+ , + , ■ 



, , a' '2am a"S,ae 



e = - m + H ^ „ . &c. = &c. 



Sffl 2a 



He remarks that tliese errors are not independent of one another, as all depend on the sino-le 



quantity 2ae, which may be eliminated between any two of the last-written equations : but 



that there is one case in which they are independent of one another, namely, when we assume 



2a e = 0, which of course leads to the method of least squares, and that in this case, as we 



shall have 



a 2 am 

 e = - m + ^ „ &c. = &c. 



2a"^ 



each error is determined by " the quantities of its own experiment." But this reasonino- is 

 perfectly inconclusive. In the case supposed, e e &c. are as much connected together as in 

 any other, as may be shown by eliminating 2am between the equations 



a2a?» , , o'2a?n 



e = - m + ^ ., , e = - m + ^ ., &c. = &c. ; 



2 a 2 a" 



and besides, apart from any mathematical reasoning, it is clear that as if we know one error 



we know all, so also if we assign any value to one, we have in effect assigned values to all, 



whether we use the method of least squares or any other. 



Moreover, e is not determined by the quantities of its own experiment alone, since 2am 



involves the results of all the experiments; there is no difference between this and the general 



case, except that 2ae has ceased to appear in the equations. But suppose we multiplied the 



equations of condition by any function of a, we might deduce the following values of .v and e : 



20a . e 2d)a . m 



X = — 1 



2a . (ha 2a . (jta 



a'S.cba.m a'S.cba.e 

 e= -m + -~-- — + -; ^ ^ , 

 2a . (pa 2a . (pa 



a''S,(pa.m a'S.cpa.e 



2a . (pa 2a . (pa 



Mr. Ivory's reasoning would apply word for word as before, and would show that the best 

 mode of combining the equations of condition was to employ the factors (pa, (pa', &c. whatever be 

 the form of (p. As it thus would serve to establish, at least apparently, an infinity of contra- 

 dictory results, the inference is that in no case has it any validity. 



I have now completed, though in an imperfect manner, the design indicated at the outset of 

 this paper, namely, to give an account of the different modes in wliich the subject has been 

 treated, and to .simplify the analytical investigations. If I have succeeded in doing this, the pre- 

 sent communication may tend to make a very curious subject more accessible than it has hitherto 

 been. 



Vol. VIII. Part II. Fi- 



